- PANORAMIC IMAGE PROJECTIONS -

An image projection occurs whenever a flat image is mapped onto a curved
surface, or vice versa, and is particularly common in panoramic photography.
A projection is performed when a cartographer maps a spherical globe of the
earth onto a flat piece of paper, for example. Since the entire field
of view around us can be thought of as the surface of a sphere (for all
viewing angles), a similar spherical to 2-D projection is required for photographs
which are to be printed.

Narrow Angle of View
(grid remains nearly square)

Wider Angle of View
(grid is highly distorted)

For small viewing angles, it is relatively easy to distort this into an image
on a flat piece of paper since this viewing arc is relatively flat.
Some distortion is inevitable when trying to map a spherical image onto a flat
surface, therefore each projection type only tries to minimize one type of distortion
at the expense of others. As the viewing angle increases, the viewing
arc becomes more curved, and thus the difference between panorama projection
types becomes more pronounced. When to use each projection depends largely
on the subject matter and application; here we focus on a few which are most
commonly encountered in digital photography. Many of the projection types
discussed in this tutorial are selectable as an output format for several panoramic
software packages; PTAssembler allows selection of all those which are listed.

IMAGE PROJECTION TYPES IN PHOTOGRAPHY

Grid representing sphere of vision (if standing
at center)

Flattened Sphere:

Choose a Projection Type:

Equirectangular (100% Coverage)

Rectilinear

Cylindrical

Mercator

Fisheye

Sinusoidal

Stereographic

If all the above image projection types seem a bit daunting, try to first
just read and understand the distinction between rectilinear and cylindrical
(shown in bold), as these are the ones which are most widely used when photo
stitching digital panoramas.

Equirectangular image projections map the latitude and longitude
coordinates of a spherical globe directly onto horizontal and vertical coordinates
of a grid, where this grid is roughly twice as wide as it is tall. Horizontal
stretching therefore increases further from the poles, with the north and south
poles being stretched across the entire upper and lower edges of the flattened
grid. Equirectangular projections can show the entire vertical and horizontal
angle of view up to 360 degrees.

Cylindrical image projections are similar to equirectangular,
except it also vertically stretches objects as they get closer to the north
and south poles, with infinite vertical stretching occurring at the poles (therefore
no horizontal line is shown at the top and bottom of this flattened grid).
It is for this reason that cylindrical projections are also not suitable for
images with a very large vertical angle of view. Cylindrical projections
are also the standard type rendered by traditional panoramic film cameras with
a swing lens. Cylindrical projections maintain more accurate relative
sizes of objects than rectilinear projections, however this is done at the expense
of rendering lines parallel to the viewer's line of sight as being curved (even
though these would otherwise appear straight).

Rectilinear image projections have the primary advantage that
they map all straight lines in three-dimensional space to straight lines on
the flattened two-dimensional grid. This projection type is what most
ordinary wide angle lenses aim to produce, so this is perhaps the projection
with which we are most familiar. Its primary disadvantage is that it can
greatly exaggerate perspective as the angle of view increases, leading to objects
appearing skewed at the edges of the frame. It is for this reason that
rectilinear projections are generally not recommended for angles of view much
greater than 120 degrees.

Fisheye image projections aim to create a flattened grid where
the distance from the center of this grid is roughly proportional to actual
viewing angle, yielding an image which would look similar to the reflection
off of a metallic sphere. These are generally not used as an output format
for panoramic photography, but may instead represent the input images when the
camera lens type being used for photo stitching is a fisheye lens. Fisheye
projections are also limited to vertical and horizontal angles of view of 180
degrees or less, yielding an image which fits within a circle. This would
be characterized by (otherwise straight) lines becoming progressively more curved
the further they get from the center of the image grid. A camera with
a fisheye lens is extremely useful when creating panoramas that encompass the
entire sphere of vision, since these often require stitching just a few input
photographs.

Mercator image projections are most closely related to the
cylindrical and equirectangular projection types; mercator represents a compromise
between these two types, providing for less vertical stretching and a greater
usable vertical angle of view than cylindrical, but with more line curvature.
This projection is perhaps the most recognizable from its use in flat maps of
the earth. Here we also note that an alternative form of this projection
(the transverse mercator) may be used for very tall vertical panoramas.

Sinusoidal image projections aim to maintain equal areas throughout
all grid sections. If flattening the globe of an earth, one can imagine
that this projection could be rolled back up again to form a sphere with the
same area and shape as the original. The equal area characteristic is
useful because if recording a spherical image in 2-D, it maintains the same
horizontal and vertical resolution throughout the image. This projection
is similar to the fisheye and stereographic types, except that it maintains
perfectly horizontal latitude lines from the original sphere.

Stereographic image projections are very similar to fisheye
projections, except that it maintains a better sense of perspective by progressively
stretching objects away from the point of perspective. This perspective-exaggerating
characteristic is somewhat similar to that yielded by the rectilinear projection,
though certainly less pronounced.

EXAMPLES: WIDE HORIZONTAL FIELD OF VIEW

How do the above image projections actually influence a panoramic photograph?
The following series of photographs are used to visualize the difference between
two projection types most often encountered in photo stitching software: rectilinear
and cylindrical projections. These are designed to show only distortion
differences for a wide horizontal angle of view; vertical panoramas are used
later on to illustrate differences in vertical distortion between other projection
types.

The first example demonstrates how a rectilinear image projection would be
rendered in a photo stitch of the above three photographs.

Note the extreme distortion near the edges of the angle of view, in addition
to the dramatic loss in resolution due to image stretching. The next image
demonstrates how the highly distorted image above would appear if it were cropped
to contain just a 120 degree horizontal angle of view.

Here we see that this cropped rectilinear projection yields a very suitable
look, since all straight architectural lines are rendered straight in the stitched
photograph. On the other hand, this is done at the expense of maintaining
the relative size of objects throughout the angle of view; objects toward the
edge of the angle of view (far left and right) are significantly enlarged compared
to those at the center (tower with doorway at base).

The next example demonstrates how the stitched photographs would appear using
a cylindrical projection. Cylindrical projections also have the advantage
of producing stitched photographs with relatively even resolution throughout,
and also require minimal cropping of empty space. Additionally, the difference
between cylindrical and equirectangular is negligible for photographs which
do not have extreme vertical angles of view (such as the example below).

EXAMPLES: TALL VERTICAL FIELD OF VIEW

The following examples illustrate the difference between projection types
for a vertical panorama (with a very large vertical field of view). This
gives a chance to visualize the difference between the equirectangular, cylindrical
and mercator projections, even though these would have appeared the same in
the previous example (with a wide horizontal angle of view).

Cylindrical

Mercator

Equirectangular

Note: The point of perspective for this panorama was set
as the base of the tower, therefore the effective vertical angle of view
looks as if there were a 140 degrees field of view in total (if the perspective
point were at the halfway height).

This large vertical angle of view allows us to clearly see how
each of these image projections differ in their degree of vertical
stretching/compression. The equirectangular projection compresses
vertical perspective so greatly that one arguably loses the sense
of extreme height that this tower gives in person. For this
reason, equirectangular is only recommended when absolutely necessary
(such as in stitched photographs with both an extreme vertical and
horizontal field of view).

The three projections above aim to maintain nearly straight vertical
lines; the transverse mercator projection to the right sacrifices
some curvature for a (subjectively) more realistic perspective.
This projection type is often used for panoramas with extreme vertical
angles of view. Also note how this projection closely mimics
the look of each of the individual source photographs.

The difference between rectilinear and cylindrical is barely noticeable
for this narrow horizontal angle of view, so the rectilinear projection
was not included.

Transverse Mercator

PANORAMIC FIELD OF VIEW CALCULATORS

The following calculator can be used to estimate your camera's vertical and
horizontal angles of view for different lens focal lengths, which can help in
assessing which projection type would be most suitable.

Panoramic Field of View Calculator

Lens Focal Length:

mm

Horizontal Size

photo(s)

Vertical Size

photo(s)

Camera Orientation:

Percent Overlap:

%

Camera Type:

Field of View:
x
(horizontal
x vertical)

Note: Calculators not intended for use in extreme macro
photography. The above results are only approximate, since the
angle of view is actually also influenced (to a lesser degree) by the
focusing distance. Additionally, field of view estimate assumes
that the lens performs a perfect rectilinear image projection; lenses
with large barrel or pincushion distortion may yield slightly different
results.

The next calculator estimates how many photos are required to encompass a
360 degree horizontal field of view, given the input settings of: focal length,
camera orientation, photo overlap and
digital camera sensor size.

360° Panorama Calculator

Lens Focal Length:

mm

Camera Orientation:

Percent Overlap:

%

Camera Type:

Required Number
of Horizontal Photos:

Note: CF = crop factor, which describes the relative
width of the camera sensor compared to a 35 mm camera. For a background
reading, please visit the tutorial on
digital camera sensor sizes.

For a summary of when to consider each projection type, please refer to the
table below:

Projection Type

Field of View Recommendations

Straight Lines?

Horizontal

Vertical

Horizontal

Vertical

Rectilinear

<120°

<120°

YES

YES

Cylindrical

~120-360°

<120°

NO

YES

Mercator

~120-360°

<150°

NO

YES

Equirectangular

~120-360°

120-180°

NO

YES

Fisheye

<180°

<180°

NO

NO

Note: All straight line considerations exclude the centermost
horizontal and vertical lines, and fields of view assume that the point
of perspective is located at the center of this angle.